Mathematical Sciences: Analysis of Patterns and Dynamics of Nonlinear Dissipative Systems

本文第一部分研究了具有无界化学非线性和任意Lewis、Prandtl和Rayleigh数的有限和无限垂直条上二维Boussinesq燃烧系统前解的整体存在性和渐近性。数值方法将用于计算前速度和控制混沌火焰前沿。第二部分讨论了金兹堡-朗道方程涡旋解的稳定性、不稳定性及其渐近分析。第三部分讨论了普适吸引子的维限、解的定性性质、使用简化方程（如斯威夫特-霍恩伯格方程）的近似以及麦克斯韦-布洛赫二能级激光系统的数值模拟。上述提出的工作是基于在反应流动动力学和激光中产生的问题。了解反应流具有重要的现实意义，涉及工业化学加工、能源消耗以及环境的生物技术修复。在设计内燃机或进行工业化学过程时，一个紧迫的问题是提高燃料燃烧的效率，从而最大限度地减少废气排放，减少空气污染。已知混合充分且搅拌强烈的反应流体具有更大的有效接触面积和更高的反应效率。另一方面，搅拌后的流体表现为不规则和混沌，不易控制。这促使我们研究不规则反应流的性质，以便利用它们来满足我们的需要。这里提出的数学模型是进行这种研究的起点，其中分析和数值方法都可以应用于推进我们的理解。此外，我们开发的技术对其他问题也非常有用，例如污染含水层的生物修复。由于化学物质与细菌生长的相互作用，这里也存在类似的不规则或混沌现象。混沌现象也出现在光学系统中，人们试图产生高功率激光输出，用于大规模并行光学计算和处理、图像存储和高功率能量源。能够实现激光输出的相干性和控制是至关重要的。在数学上，这相当于研究激光系统的稳定性和不稳定性的特解，分析结构和寻找良好的近似吸引子。本文提出的两能级激光模型是实现这一目标的第一步。在两能级模型的系统研究中所取得的进展和发展的方法将极大地帮助我们在未来接近更复杂但更有吸引力的半导体激光系统。

Abstract Xin The first part of the proposed research concerns the global existence and asymptotics of front solutions to the two-dimensional Boussinesq combustion system on finite and infinite vertical strips with unbounded chemical nonlinearities, and arbitrary Lewis, Prandtl and Rayleigh numbers. Numerical methods will be implemented for calculating front speeds and controling chaotic flame fronts. The second part concerns the stability, instability of vortex solutions to Ginzburg-Landau equations and related asymptotic analysis. The third part concerns the dimensional bounds of the universal attractors, qualitative properties of solutions, approximation using reduced equations (such as Swift-Hohenberg like equations), and numerical simulation of the Maxwell-Bloch two level laser systems. The above proposed works are based on problems arising in dynamics of reacting flows, and lasers. Understanding reacting flows is of tremendous practical importance, and is related to industrial chemical processing, energy consumption, as well as biotechnological remediation of environment. In designing internal combustion engines or conducting industrial chemical processes, one of the immediate concerns is to increase the efficiency of fuel burning, and so minimize the waste gas output and reduce air pollution. It is known that well-mixed and strongly stirred reacting fluids have more effective contact area and their reaction efficiency is much higher. On the other hand, stirred fluids behave in an irregular and chaotic manner, and are not easy to control. This motivates us to study the properties of the irregular reacting flows in order to utilize them for our needs. Mathematical models proposed here serve as a starting point for pursuing such an investigation where both analytical and numerical methods can be applied to advance our understanding. Moreover, the techniques we develop can be very useful to other problems, such as bioremediation of polluted aquifer. T here similar irregular or chaotic phenomena exist due to the interaction of chemical species with bacteria growth. Chaotic phenomena also appear in optical systems where people try to produce high power laser output for massively parallel optical computing and processing, image storage, and high power energy sources. It is essential to be able to achieve coherence and control of the laser output. Mathematically, this amounts to studying stability and instability of the special solutions of the laser system, analyzing the structures and finding good approximations of attractors. The two level laser model we propose here is the first step towards this goal. The progress made and the methodology developed in the systematic study of the two level model will greatly help us approach more complicated yet more appealing semiconductor laser systems in the future.